To simplify the given expression, we can break it down into individual terms and then simplify each term separately.

The general form of each term is 1/n(1+2+3+...+n), where n ranges from 1 to 20.

Let's calculate each term separately:

For n = 1:
1/1(1) = 1

For n = 2:
1/2(1+2) = 1/2(3) = 3/2

For n = 3:
1/3(1+2+3) = 1/3(6) = 2

Continuing this pattern, we can calculate each term up to n = 20.

For n = 4:
1/4(1+2+3+4) = 1/4(10) = 5/2

For n = 5:
1/5(1+2+3+4+5) = 1/5(15) = 3

For n = 6:
1/6(1+2+3+4+5+6) = 1/6(21) = 7/2

Continuing this pattern, we can calculate each term up to n = 20.

For n = 20:
1/20(1+2+3+...+20) = 1/20(210) = 21/2

Now, we can add up all the calculated terms:

b = 1 + 3/2 + 2 + 5/2 + 3 + 7/2 + ... + 21/2

To simplify this expression, we can combine like terms:

b = (1 + 2 + 3 + ... + 20) + (3/2 + 5/2 + 7/2 + ... + 21/2)

The sum of the numbers from 1 to 20 can be calculated using the formula for the sum of an arithmetic series:

Sum = (n/2)(first term + last term)
Sum = (20/2)(1 + 20)
Sum = 10(21)
Sum = 210

Now, let's simplify the second part of the expression:

(3/2 + 5/2 + 7/2 + ... + 21/2)

This is an arithmetic series with a common difference of 2 and a first term of 3/2. The last term can be calculated using the formula for the nth term of an arithmetic series:

nth term = first term + (n-1) * common difference
21/2 = 3/2 + (n-1) * 2

Simplifying this equation, we get:
21/2 = 3/2 + 2n - 2
21/2 - 3/2 + 2 = 2n
18/2 + 2 = 2n
9 + 2 = 2n
11 = 2n
n = 11/2

Now, we can calculate the sum of this arithmetic series using the formula:

Sum = (n/2)(first term + last term)
Sum = (11/2)(3/2 + 21/2)
Sum = (11/2)(24/2)
Sum = (11/2)(12)
Sum = 66

Finally, we can add the two sums together:

b = 210 + 66
b = 276

Therefore, the value of b is 276.